Computing Fundamentals 1 Lecture 0

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Computing Fundamentals 1Lecture 0

• Lecturer Patrick Browne
• http//www.comp.dit.ie/pbrowne/Compfund1/compfun1.
  htm
• Room K308
• Based on Chapter 0.
• A Logical approach to Discrete Math
• By David Gries and Fred B. Schneider

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Leibniz Logician, Mathematician

• It is unworthy of excellent men to lose hours
  like slaves in the labour of calculations which
  could safely be relegated to anyone else if
  machines were used. Leibniz 1646-1716.
• The ideas of Leibniz are extensively used in the
  course text.

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Computing Fundamentals 1

• This module presents theoretical aspects of
  computer science which are necessary to support
  and enhance other modules on the course. In
  particular the topics covered on this module will
  be required in computer technology, database,
  software engineering, programming, and
  algorithms.

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Computing Fundamentals 1

• Logic Propositional calculus, truth tables,
  logical equivalence, logical argument, predicate
  calculus, simple proofs.
• Set Theory Algebra of sets, power sets,
  cardinality, Venn diagrams, programming using
  sets.
• Relations Types, representations, equivalence,
  partial order, relational database theory.
• Functions The graph of a function, properties,
  composition, functions in programming languages.
• Boolean Algebra Basic laws, simplification of
  expressions, application to switching circuits.
• Number Systems Binary, octal, decimal,
  hexadecimal, simple binary arithmetic.
• Supporting software The above topics will be
  supported by software tools including functional
  and logic based languages.

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Computing Fundamentals 1

• The aim of this module is to provide the student
  with the theoretical foundations for other
  modules on the programme.

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Computing Fundamentals 1

• Main Reference
• A Logical approach to Discrete Math
• By David Gries and Fred B. Schneider
• Publisher Springer 1 edition (1993)
• ISBN 0387941150

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Computing Fundamentals 1

• Recommended Reading
• Seymour Lipschutz, Essential Computer
  Mathematics, Schaum's Outline series, 1987, ISBN
  0-07-0379990-4.
• Winfred Karl Grassmann and Jean-Paul Tremblay ,
  1996, Logic and Discrete Mathematics A Computer
  Science Perspective, Prentice Hall 1996, ISBN
  0-13-501206-6.
• Seymour Lipschutz and Marc Lars Lipson, 1997,
  Discrete Mathematics, Schaum's Outline series,
  ISBN 0-07-038045-7.  

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Mathematics Logic

• Mathematics can be used to represent, or model
  the world. Math provides a precise, concise
  representation of quantities and relationships.
  As Information Scientists we are required to
  reason correctly to reach conclusions, and hence
  we need a knowledge of formal logic.

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Formalizing Common Sense

• The following pair of signs are displayed at the
  foot of an escalator. What do they mean? They
  have exactly the grammatical structure.

Dogs Must Be Carried
Shoes Must Be Worn
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Syntax and Semantics

• What do the following symbols mean?
• The number six
• VI
• 3 3
• The smallest whole number greater that five.
• The integer successor of five.
• The positive square root of 36.

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Syntax and Semantics

• We need to separate symbols from meaning.
• What do the following symbols mean?

27 128 26 64 25 32 24 16 23 8 22
4 21 2 20 1
102 100 101 10 100 1
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Computing Powers in CafeOBJ

• mod! POWER
• protecting(NAT)
• op __ Nat Nat -gt Nat
• vars X I Nat
• eq X 0 s 0 . s is successor
  function
• eq X I X (X ( p I)) .
• -- Open the module and reduce expression
• open POWER
• red (2 7) (2 5) (2 4) (2 1) (2
  0) .
• (179)NzNat

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Syntax and Semantics
Syntax refers to the structure of expressions, or
rules for putting symbols together to form
expressions. Semantics refers to the meaning of
expressions or how they are evaluated.
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Syntax and Semantics
Syntax refers to the structure of expressions, or
rules for putting symbols together to form
expressions. Semantics refers to the meaning of
expressions or how they are evaluated.
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Logic

• Logic is the glue that binds together methods of
  reasoning, in all domains. Proof methods have
  their basis in formal logic. An interpretation
  assigns meaning to the operators, constants, and
  variables of a logic (model theory). From Gries
  Schneider1
• Logic is the study of reasoning it is
  specifically concerned with whether reasoning is
  correct. Logic focuses on the relationship among
  statements as opposed to the content of any
  particular statement. From Johnsonbaugh2
• Logic as Theory of Science People can hold
  scientific theories. A scientific theory consists
  of a multiplicity of acts of knowing, of
  verifying and falsifying, validating and
  calculating. Logic is a theory which seeks to
  determine the conditions which must be satisfied
  by a collection of acts if it is to count as a
  science. Logic is a theory of science. Barry
  Smith3

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Logic and Language

• Logic takes into account syntactically
  well-formed sentences and studies whether they
  are semantically correct.
• There are sentences that are syntactically
  correct but not from the semantic point of view.
  For instance "The big house is green" is both
  syntactically and semantically correct (provided
  the house under consideration is green), while
  "The big house is small" is contradictory as a
  sentence and thus semantically incorrect.

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Logic and Language

• Keep in mind the distinction between formal logic
  on this course and the intuitive everyday logic
  (informal)
• Further we often use informal logic to reason
  about formal logic. For example, we can say the a
  particular logic is not suited to a particular
  task.

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Logic and Language

• We will start with a logic called equational
  logic
• We will use the CafeOBJ language, which is based
  on equational logic. We will use CafeOBJ as a
  functional programming language and as a theorem
  prover.
• The role of CafeOBJ on this course is to provide
  a logically based language that can be used to
  represent the mathematical concepts such as logic
  itself, sets, functions, relations and even
  programs (considering programs as mathematical
  objects). At the end of the course you should
  be able to read and understand small CafeOBJ
  proofs and programs.

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Mathematical Models

• Mathematical models are more concise, precise,
  and rigorous than informal English descriptions.
• A mathematical model allows us compute solutions.
• A mathematical model allows us to reason about
  the problems in a mechanical way. This allows us
  to
• manipulate expressions
• prove properties from and about expressions
• obtain new results from know facts or expressions

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Mathematical Models

• There are rules that allow is to perform
  syntactic manipulations without regards for
  semantics.
• Rigour does not mean complexity.
• Rigour is defined as strict precision or
  exactness.
• Rigour usually leads to simplicity of proof and
  extensibility.
• With a mathematical based language, like CafeOBJ,
  one can model, reason, prove, and compute
• A value is, a computation does. A value has value
  type, a computation has computation type.

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Equational logic

• Equational logic (EL) can be used as a tool to
  reason about systems. EL is based on equality and
  Leibnizs rule for substituting equals for
  equals.
• You should be familiar with EL from school. For
  example. Mary has twice as many apples as John,
  can be written as
• m 2 j
• There are many values of m and j that make the
  equation true (e.g. m4 and j2, m6 and j3).

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Equational logic

• Mary has twice as many apples as John. Mary
  throws half her apples away, because they were
  rotten, and John eats one of his. Mary still has
  twice as many apples as John. How many apples did
  Mary and John have initially?
• (0.1) m 2 j and
• m/2 2 (j 1)
• There are now fewer values of m and j that make
  these two equations true (m4 and j2).

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The Apple0 problem 1

• mod APPLES0
• Apples
• var M Apples -- Mary's apples
• var J Apples -- Johns apples
• ops 0 1 2 -gt Apples
• ops quo Apples Apples -gt Apples
• eq eq1 M J 2 .
• eq eq2 M quo 2 2 (J 1) .
• The above specification represents the mini-world
  of Mary and Johns apples. As it stands it does
  not compute a solution.
• we have symbol
• representing multiplication operation
• quo representing division operation (quotient)
• 0, 1, 2 representing numbers
• M,N representing number of Apples
• But these as yet do not have their normal
  computable interpretation. You could say that at
  the moment the symbols have an intended
  interpretation. Our priority is initially
  representation not computation.
• See Apples1.mod and Apples2.mod on web page.

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The Apple problem 1
A quotient is the result of a division. Dividing
6 by 3, the quotient is 2

• mod! APPLES1
• Apples
• var M Apples -- Mary's apples
• var J Apples -- Johns apples
• ops 0 1 2 -gt Apples
• op _quo_ Apples Apples -gt Apples assoc
• op __ Apples Apples -gt Apples assoc
• op _-_ Apples Apples -gt Apples assoc
• eq eq1 2 J quo 2 2 (J - 1) .
• eq eq2 2 (J - 1) 2 J - 2 .
• eq eq3 2 J quo 2 J . -- what this
  equation mean?
• To test if an equation, say eq1, is true
• open APPLES1
• reduce 2 J quo 2 2 (J - 1) . true

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The Computing the Apple problem

• mod! APPLES2
• pr(INT)
• op solution Int -gt Int
• var M Int -- Mary's apples
• var J Int -- Johns apples
• eq solution(J)
• if (2 J quo 2 2 (J - 1)) then J
• else
• solution(J 1) fi .
• Marys apples not used to compute J.

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Pre-condition, Post-condition

• A pre-condition of a program (or algorithm)
  statement is an assertion about the program
  variables in a state in which the statement must
  be executed, and a post-condition is an assertion
  about the state in which it may terminate. A
  reasonable precondition for the last program is
  that the first guess should be equal or greater
  than zero (no negative number of apples)

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More Examples

• In English we wish approximate
• In Math
• Let b approximate (therefore b2 approx. n)
• Precondition 0 lt n
• Postcondition b2 lt n lt (b1)2
• The postcondition that n computes the largest
  integer that is at most
• These conditions do not tell us how to compute.
• With maths can model, reason, and compute

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More Examples

• Here is an equation in CafeOBJ that will allow us
  to compute an approximation of the square root of
  a number.
• eq approx(B, N)
• if ((B B lt N) and
• (N lt (B 1) (B 1)) )
• then B
• else
• approx(B 1, N) fi .
• (see ROOTN1.MOD on web page)

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Systematic Syntactic manipulation

• From emc2 show e/c2m
• Precondition 0ltc2
• e mc2
• ltDivide both sides by c2gt
• e/c2 (mc2)/c2
• ltAssociativity of / and gt
• e/c2 m(c2/c2)
• lt(c2/c2) 1 gt
• e/c2 m1
• ltm1 m gt
• e/c2m
• Transitivity if ab and bc then we can conclude
  that ac
• Equality is transitive, hence we can conclude
  emc2 is equivalent e/c2m

Einstein showed a massenergy equivalence. Matter
can be turned into energy, and energy into
matter. Here we focus on syntactic manipulation
disregarding the meanings of the symbols.
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Mathematical expressions

• Syntax of mathematical expressions are
  constructed from constants, variables, and
  operators
• A constant such as 123 is an expression
• A variable such as y is an expression1.
• An operator such as used in expression
• 123 y
• Brackets can be used to aggregate expressions
  e.g. 2 (3 4) .

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Mathematical expressions

• Syntax refers to the structure of expressions, or
  rules for putting symbols together to form
  expressions.
• Semantics refers to the meaning of expressions or
  how they are evaluated.
• We are interested in syntax, semantic, and how
  they are related.

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Equational logic

• State is a list of variables with associated
  values.
• Evaluation of an expression E in a state is
  performed by replacing all variables in E by
  their values in the state and then computing the
  value of the resulting expression. For example
• Expression x y 2
• State (x,5),(y,6)
• Gives 5 6 2
• Evaluates to 1

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Equational logic

• We should to distinguish between the syntactic
  manipulation of formulas and the evaluation and
  meaning of those formulas.
• See
• http//www.cs.cornell.edu/gries/logic/Equational.h
  tml
• http//homepages.feis.herts.ac.uk/comqejb/algspec
  /node8.html
• http//scom.hud.ac.uk/scomtlm/book/node275.html

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Equational logic (Wikipedia)1

• Equational logic (EQ) provides a set of
  domain-independent rules. The rules provide a
  sound logic for numerical domains and many other
  kinds of mathematical structures such as sets.
  The reasoning rules for EQ.
• Reflexivity. t t. That is, any equation whose
  two sides are the same term t is a law2.
• Symmetry. From s t infer t s. That is, the
  two sides of a law may be interchanged.
  Intuitively one attaches no importance to which
  side of an equation a term comes from.
• Transitivity. A chain s t u of two laws
  yields the law s u.3
• Substitution. Given two laws and a variable, each
  occurrence of that variable in the first law may
  be replaced by one or the other side of the
  second law.4

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Programming Logic

• The basic components of current languages are
• Data types e.g. Integers, String, Polygon.
• Variables to refer to data types e.g. a 2
• Operations on data types e.g. area(polygon)
• Control structures e.g. sequence, iteration, and
  conditions.
• Logic is an important part of programming, but it
  is often implicit and external to the language.
  Some languages like Prolog or SQL are quite close
  to logic.
• In many cases a CafeOBJ program has a one-to-one
  relation with is mathematical or logical meaning.

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CafeOBJ

• The CafeOBJ system and documentation can be
  downloaded from http//www.ldl.jaist.ac.jp/cafeob
  j/
• You should download the latest version.
• The site contains installation instructions for
  Linux, Mac OS X, and Windows
• There are excellent CafeOBJ notes by Kokichi
  Futatsugi at
• http//www.jaist.ac.jp/kokichi/class/i613-0712/
• Kazuhiro OGATA has written an excellent set of
  notes on functional programming using CafeOBJ
• http//www.jaist.ac.jp/ogata/lecture/i217/
• Takahiro Seino has written an excelent
  introduction to CafeOBJ as a theorem prover.
• http//www.jaist.ac.jp/t-seino/lectures/cafeobj-i
  ntro/en/index.html
• The CafeOBJ manual is useful but a bit detailed
  and technical. User-friendly tutorials will be
  provided at the labs.

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Staring CafeOBJ

• After installing CafeOBJ there are typically four
  things that you need to do to work on a file
• 1. In a command prompt, start CafeOBJ, type
  CafeOBJ.bat
• 2. Set CafeOBJ to point to files in your current
  working directory (or folder), type
• cd yourDir
• 3. Load the file you wish to work on
• in workingFile
• 4)Open the module you wish to working with
• open ModuleName